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arXiv · 2201.12942

The road problem and homomorphisms of directed graphs

Sophie MacDonald
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Abstract

We make progress on a generalization of the road (colouring) problem. The road problem was posed by Adler-Goodwyn-Weiss and solved by Trahtman. The generalization was posed, and solved in certain special cases, by Ashley-Marcus-Tuncel. We resolve two new families of cases, of which one generalizes the road problem and follows Trahtman's solution, and the other generalizes a result of Ashley-Marcus-Tuncel with a proof quite different from theirs. Along the way, we prove a universal property for the fiber product of certain graph homomorphisms, which may be of independent interest. We provide polynomial-time algorithms for relevant constructions and decision problems.

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Sophie MacDonald. 2022-01-31. The road problem and homomorphisms of directed graphs. https://doi.org/10.1016/j.tcs.2023.113981

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